PRL 113, 144101 (2014)
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PHYSICAL REVIEW LETTERS
Optimal Synchronization of Complex Networks
Per Sebastian Skardal,1,2,* Dane Taylor,2,3,4,† and Jie Sun5,‡
1
Departament d’Enginyeria Informatica i Matemátiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
2
Department of Applied Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309, USA
3
Statistical and Applied Mathematical Sciences Institute, Research Triangle Park, North Carolina 27709, USA
4
Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599, USA
5
Department of Mathematics, Clarkson University, Potsdam, New York 13699, USA
(Received 28 February 2014; revised manuscript received 19 June 2014; published 30 September 2014)
We study optimal synchronization in networks of heterogeneous phase oscillators. Our main result is the
derivation of a synchrony alignment function that encodes the interplay between network structure and
oscillators’ frequencies and that can be readily optimized. We highlight its utility in two general problems:
constrained frequency allocation and network design. In general, we find that synchronization is promoted
by strong alignments between frequencies and the dominant Laplacian eigenvectors, as well as a matching
between the heterogeneity of frequencies and network structure.
DOI: 10.1103/PhysRevLett.113.144101
PACS numbers: 05.45.Xt, 89.75.Hc
A central goal of complexity theory is to understand
the emergence of collective behavior in large ensembles
of interacting dynamical systems. Synchronization of
network-coupled oscillators has served as a paradigm for
understanding such emergence [1–4], where examples arise
in nature (e.g., flashing of fireflies [5] and cardiac pacemaker cells [6]), engineering (e.g., power grid [7] and
bridge oscillations [8]), and at their intersection (e.g.,
synthetic cell engineering [9]). We consider the dynamics
of N network-coupled phase oscillators θi for i ¼ 1; …; N,
whose evolution is governed by
θ_ i ¼ ωi þ K
N
X
Aij Hðθj − θi Þ:
ð1Þ
j¼1
Here ωi is the natural frequency of oscillator i, K > 0 is
the coupling strength, [Aij ] is a network adjacency
matrix, and H is a 2π-periodic coupling function [10].
We treat HðθÞ with full generality provided H0 ð0Þ > 0. The
choices HðθÞ ¼ sinðθÞ and HðθÞ ¼ sinðθ − αÞ with the
phase-lag parameter α ∈ ð−π=2; π=2Þ yield the classical
Kuramoto [10] and Sakaguchi-Kuramoto models [11],
respectively.
Considerable research has shown that the underlying
structure of a network plays a crucial role in determining
synchronization [12–20], yet the precise relationship
between the dynamical and structural properties of a
network and its synchronization remains not fully understood. One unanswered question is, given an objective
measure of synchronization, how can synchronization be
optimized? One application lies in synchronizing the power
grid [21], where sources and loads can be modeled as
oscillators with different frequencies. To this end, we ask:
what structural and/or dynamical properties should be
present to optimize synchronization?
0031-9007=14=113(14)=144101(5)
We measure the degree of synchronization of an ensemble of oscillators using the Kuramoto order parameter
reiψ ¼ ð1=NÞ
N
X
eiθj :
ð2Þ
j¼1
Here reiψ denotes the phases’ centroid on the complex unit
circle, with the magnitude r ranging from 0 (incoherence)
to 1 (perfect synchronization) [10]. In general, the question
of optimization (maximizing r) is challenging due to the
fact that the macroscopic dynamics depends on both the
natural frequencies and the network structure. To quantify
the interplay between node dynamics and network structure, we derive directly from Eqs. (1) and (2) a synchrony
alignment function, which is an objective measure of
synchronization and which can be used to systematically
optimize a network’s synchronization. We highlight this
result by addressing two classes of optimization problems,
which can be easily adapted to a wide range of applications.
The first is constrained frequency allocation, where given a
fixed network topology, optimal frequencies are chosen.
The second is network design, where given a fixed set of
frequencies, an optimal network structure is found. We next
present the derivation of the synchrony alignment function.
No assumptions are made about the frequencies or the
network aside from the network being connected and
undirected.
We begin by considering the dynamics of Eq. (1) in
the strong coupling regime where r ≲ 1, which may
typically be obtained by either increasing the coupling
strength or decreasing the heterogeneity of the frequencies. In this regime the oscillators are entrained in a tight
cluster such that θi ≈ θj for all (i, j) pairs. Expanding
Eq. (1) yields
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PRL 113, 144101 (2014)
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PHYSICAL REVIEW LETTERS
θ_ i ≈ ω~ i − KH0 ð0Þ
N
X
(a)
Lij θj ;
(b)
ð3Þ
j¼1
P
~ i ¼ ωi þ KHð0Þdi , di ¼ Nj¼1 Aij is the degree of
where ω
node i, and [Lij ] is the Laplacian matrix whose entries are
defined as Lij ¼ δij di − Aij .
The following spectral properties of the Laplacian are
essential to our analysis. First, since the network is
connected and undirected, all eigenvalues are real and
can be ordered 0 ¼ λ1 < λ2 ≤ ≤ λN−1 ≤ λN . Second,
the normalized eigenvectors fvi gNi¼1 form an orthonormal
basis for RN. Furthermore, the eigenvector associated with
λ1 ¼ 0 is v1 ¼ 1, which corresponds to the synchronization
manifold.
Inspecting Eq. (3), we find that if a steady-state solution
θ exists after entering the rotating frame θi ↦ θi þ Ωt
~ it is given by
where Ω is the mean hωi,
0
~
θ ¼ L† ω=KH
ð0Þ;
ð4Þ
P
j jT
is the pseudoinverse of L [22].
where L† ¼ Nj¼2 λ−1
j v v
Under the approximation in Eq. (3), the steady-state
solution is expected to be linearly stable since the
Jacobian matrix is approximately given by −KH0 ð0ÞL
and has nonpositive eigenvalues. We next consider the
order parameter given θ . First, with a suitable shift in
initial conditions the average phase can be set to zero,
implying that the sum in Eq. (2) is real. Furthermore, in the
strongly synchronized regime all phases are tightly packed
about ψ ¼ 0, thus jθj j ≪ 1 for all j. Expanding Eq. (2)
yields
r ≈ 1 − jjθ jj2 =2N:
ð5Þ
Finally, by the spectral decomposition of the pseudoinverse
L† and writing the norm in Eq. (5) by taking the inner
product of θ in Eq. (4), we obtain
~ LÞ=2K 2 H02 ð0Þ;
r ≈ 1 − Jðω;
ð6Þ
for which we define the synchrony alignment function
~ LÞ ¼ ð1=NÞ
Jðω;
N
X
j ~ 2
λ−2
:
j hv ; ωi
ð7Þ
j¼2
~ LÞ is our main theoretical result as
The derivation of Jðω;
its minimization corresponds to the maximization of the
order parameter r, which allows for the optimization of
synchronization using elementary properties of the network
(Laplacian eigenvalues and eigenvectors) and the frequen~ LÞ, we note
cies. Before exploring the optimization of Jðω;
the following interesting results. For Hð0Þ ¼ 0, it follows
~ ¼ ω, and thus optimization of Jðω; LÞ is indepenthat ω
dent of K. However, for Hð0Þ ≠ 0 perfect synchrony,
FIG. 1 (color online). Constrained frequency allocation: (a) r vs
K for optimal allocation (blue circles) compared with random
allocations drawn from normal (red triangles), uniform (green
pluses), and Laplace (orange squares) distributions. (b) r vs K for
a prechosen set of normally distributed frequencies with random
(red triangles), first-order (black crosses), and near-optimal (blue
circles) allocations. The near-optimal allocation was obtained
from 106 proposed frequency exchanges. Networks are scale free
(SF) with N ¼ 1000, γ ¼ 3, and d0 ¼ 2.
r ¼ 1, is generally not attainable unless Jðd; LÞ ¼ 0 (which
can occur if d1 ¼ d2 ¼ ¼ dN ) since in the limit K → ∞
Eq. (6) yields r ¼ 1 − Jðd; LÞH 2 ð0Þ=2H02 ð0Þ. It follows
that the existence of a strong coupling regime, r ≲ 1; and,
consequently, the approximations in our theory are valid
only when Jðd; LÞH2 ð0Þ=2H02 ð0Þ ≪ 1. Furthermore, since
~ depends on the coupling strength K, so will the
ω
~ LÞ and therefore the optimal network.
optimization of Jðω;
From now on, we will specialize to the widely used
Kuramoto model, HðθÞ¼sinðθÞ [which gives Hð0Þ ¼ 0],
although we emphasize that similar results are found for
more general coupling functions HðθÞ.
We first address constrained frequency allocation for a
fixed network. We note that by entering a rotating frame
we can without loss of generality set the mean frequency to
zero. The choice ω ¼ ½0; …; 0T trivially minimizes
Eq. (7), resulting in r ¼ 1, so we require asp
a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
first constraint
P ffi
that ω has a fixed standard deviation, σ ¼ N −1 i ω2i . By
rescaling time and the coupling strength, σ can be tuned
freely, so without loss of generality we set σ ¼ 1. To
minimize Jðω; LÞ, we first express ω as a linear
P combination of the nontrivial eigenvectors of L, ω ¼ Ni¼2 αi vi ,
PN 2
where the coefficients must satisfy
i¼2 αi ¼ N. After
inserting ω into Eq. (7) it follows that Jðω;pLÞ
ffiffiffiffi is minimized
by the choice α2 ; …; αN−1 ¼ 0 and αN ¼ N , i.e., ω ∝ vN ,
yielding r ¼ 1 − 1=2λ2N K 2 .
In Fig.
pffiffiffiffi 1(a) we compare the results of optimal allocation,
ω ¼ N vN , with several random frequency allocations by
plotting the synchronization profiles r vs K for the optimal
allocation, and those for frequencies randomly drawn from
normal, uniform, and Laplace distribution (each with
unit standard deviation). The underlying network with
N ¼ 1000 nodes was constructed using the configuration
model [23] with a scale free degree distribution PðdÞ ∝ d−γ
for γ ¼ 3 and minimum degree d0 ¼ 2. The optimal
allocation shows a large improvement over all random
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PHYSICAL REVIEW LETTERS
(a)
(b)
(c)
(d)
(e)
(f)
FIG. 2 (color online). Optimal network design: (a)–(b) r vs K
for initial (red crosses) and rewired (blue circles) networks with
normal and power-law distributed frequencies. (c)–(d) Degree
distributions of the initial (red crosses) and rewired (blue circles)
networks. (e)–(f) Illustrations for networks of N ¼ 40 and
N ¼ 36 nodes, respectively after rewiring.
allocations and is marked by a sharp transition to a strongly
synchronized state at a small coupling strength. In particular, the optimal allocation is surprisingly effective at very
small coupling strengths despite the strong coupling
assumption in the theory. We note that two mechanisms
contribute to the excellent performance of the optimal
allocation: the choice of frequencies and the nodes to which
they are assigned.
To elucidate the importance of these two different
mechanisms, we consider an additional constraint where
frequencies fωi gNi¼1 are prechosen and must be allocated
optimally on the network to minimize Jðω; LÞ. Finding the
global minimum requires an exhaustive search over all N!
possible permutations of ω, which is an unrealistic option
even for moderately sized networks. We therefore provide
two alternatives: a first-order approximation applicable for
networks in which the largest Laplacian eigenvalue λN is
well separated from the others (often the case for SF
networks [24]) and a near-optimal solution based on an
accept-reject algorithm. In particular, when the dominant
eigenvalue is well separated, λi ≪ λN for i ≠ N, an inexpensive first-order minimization of the objective function
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leads to maximizing jhvN ; ωij. This can be done simply by
finding the index permutations i1 ; …; iN and j1 ; …; jN that
place eigenvector entries in ascending order, vNi1 ≤ ≤ vNiN ,
and frequencies in ascending (or descending) order,
ωj1 ≤ ≤ ωjN (or ωj1 ≥ ≥ ωjN ). In principle both
pairings must be checked to select the best result. To find
a near-optimal allocation we begin with an initial choice ω
and construct a new vector ω0 by exchanging two randomly
chosen entries. If Jðω0 ; LÞ < Jðω; LÞ we accept ω0 , otherwise we reject it. This procedure is then repeated for a
number of proposed exchanges.
In Fig. 1(b) we compare synchronization profiles for
near-optimal, first-order, and random allocations where
frequencies are drawn from the unit normal distribution. As
expected, the near-optimal allocation yields the best results;
however, the first-order allocation also performs well,
providing an inexpensive way to improve upon purely
random allocation. These results also allow us to compare
the allocation of prechosen frequencies to freely chosen
frequencies [Fig. 1(a)]. In both cases, the transition from
incoherence to strong synchronization is sharp; however, it
occurs at a larger coupling strength (K ≈ 0.4) when
frequencies are prechosen, yielding two distinct regimes:
for small K strong synchronization is only attainable when
frequencies are freely tunable, while for larger K strong
synchronization is attainable even when the frequency set
is fixed.
Next we address the complementary problem of optimal
network design for a fixed set of frequencies. Given ω and
a fixed number of links, we look for a network that
minimizes Jðω; LÞ. As an algorithmic method for obtaining
an approximate solution, we initialize an accept-reject
algorithm with a network satisfying these constraints,
and allow it to evolve as follows. A new network with
Laplacian matrix L0 is constructed by randomly deleting a
link and introducing another between two previously
disconnected nodes. If Jðω; L0 Þ < Jðω; LÞ we accept the
new network, otherwise we reject it. This procedure is then
repeated for S proposed rewirings. In Fig. 2 we present the
results of this rewiring algorithm for two experiments. We
consider two networks: one with relatively homogeneous
frequencies drawn from a unit normal distribution (left
column), and a second with heterogeneous frequencies
drawn from a symmetric power-law distribution, gðjωjÞ ∼
jωj−3 (right column). Both networks contain N ¼ 1000
oscillators. In Figs. 2(a) and 2(b) we plot the synchronization profiles for the initial networks and the networks
obtained after 2 × 104 rewirings. In both experiments, the
rewired networks display better synchronization properties
with sharp transitions from incoherence to strong synchronization. Each experiment is initialized with a different
network topology: a SF network constructed by the
configuration model with γ ¼ 3 and d0 ¼ 2 and an
Erdős-Rényi [25] network with average degree hdi ¼ 4
are paired with the normal and power-law distributed
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(b)
FIG. 3 (color online). Eigenvector alignments: (a) r vs K for
frequency alignments ω ∝ v100 ; v200 ; …; v1000 (red to blue). (b) r
vs i for ω ∝ vi with fixed K ¼ 0.08 (blue circles), 0.25 (red
crosses), and 1 (green triangles). Each point is averaged over 50
realizations of SF networks with N ¼ 1000; γ ¼ 3 and d0 ¼ 2.
frequencies, respectively. In Figs. 2(c) and 2(d) we plot the
initial and rewired degree distributions. In both experiments
the degree distribution evolves to better match that of the
frequencies, either becoming less [Fig. 2(c)] or more
[Fig. 2(d)] heterogeneous. This is further emphasized by
the shifts in the maximal degrees d∞ , which decreases from
75 to 15 and increases from 10 to 26, respectively. This
suggests that a more heterogeneous network better synchronizes a more heterogeneous set of frequencies. To
illustrate this phenomenon, we show in Figs. 2(e) and 2(f)
networks resulting from the same experiment with fewer
nodes (N ¼ 40 and N ¼ 36, respectively). The radius of
each node is proportional to its degree and the coloring of
the node indicates its frequency from most positive (red) to
most negative (blue). Here the phenomenon is easily
observable with the emergence of network hubs in
Fig. 2(f) but not Fig. 2(e).
We now study in more detail the synchrony alignment
function given in Eq. (7). Just as aligning ω with vN
maximizes r, it follows that in the strong coupling regime,
aligning ω with other eigenvectors vi of decreasing index
yields weaker synchronization. We consider the alignments
ω ∝ vi and plot the synchronization profiles in Fig. 3(a)
for i ¼ 100; 200; …; 1000 (red to blue) averaged over 50
realizations of SF networks with parameters N ¼ 1000,
γ ¼ 3, and d0 ¼ 2. As expected, we observe weaker
synchronization with decreasing index. We also plot in
panel 3(b) r vs i for a few isolated coupling strengths:
K ¼ 0.08, 0.25, and 1, again averaged over 50 realizations.
For all three cases r tends to increase with i, provided i is
not too small. For K ¼ 0.08 the majority of alignments
yield incoherence with r, undergoing a sharp increase only
near the most dominant eigenvectors, while the increase
in r is more gradual for K ¼ 0.25 and 1. We also point out
that for very small i we observe a mild increase in r, which
we attribute to local synchronization that yield large
fluctuations in r.
Before concluding, we investigate the dynamical and
structural properties present in optimized networks.
In particular, we consider local degree-frequency and
FIG. 4 (color online). Correlations in optimized networks:
(a),(b) Frequency magnitude jωi j vs degree di and (c),(d) average
neighbor frequency hωi i vs frequency ωi for networks with
optimally chosen frequencies (green triangles), prechosen
frequencies (blue circles), and a rewired network (red crosses).
Networks are SF with N ¼ 1000, γ ¼ 3, and d0 ¼ 3. Frequencies
for the arranged and rewired cases are normally distributed.
neighboring frequency-frequency correlations. In Figs. 4(a)
and 4(b) we plot the frequency magnitude jωi j vs degree di
for, in Fig. 4(a), a network with optimally allocated
frequencies and, in Fig. 4(b), a network with prechosen
frequencies (blue circles) and a rewired network (red
crosses). Networks are SF with N ¼ 1000, γ ¼ 3, and
d0 ¼ 3 and frequencies in Fig. 4(b) are normally distributed. In each case we observe a strong positive degreefrequency correlation, indicating that the largest frequencies correspond to the network hubs. Moreover, in
Figs. 4(c) and 4(d) we plot for each node i the average
frequency of its neighboring oscillators hωii ¼
P
N
j¼1 Aij ωj =di vs ωi . The results are qualitatively similar
for each case, showing a strong negative correlation between
neighboring frequencies. These observations agree with the
findings in Refs. [26,27], where similar positive and negative
correlations were found to promote global synchronization.
We finally note that such degree-frequency correlations may
help explain the increased sharpness of transitions shown
in Figs. 1 and 2(a)–2(b), since similar correlations can lead to
abrupt and explosive transitions [17].
In this Letter we presented a synchrony alignment
function that measures the interplay between network
structure and oscillator heterogeneity and allows for a
systematic optimization of synchronization. Focusing on
Kuramoto coupling, we highlighted its utility through
numerical experiments for random networks with two
general classes of optimization problems: frequency allocation and network design. We found that synchronization
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PHYSICAL REVIEW LETTERS
is promoted by a strong alignment of the frequency vector
with the most dominant Laplacian eigenvectors and that,
relatively speaking, more (less) heterogeneous networks
better synchronize more (less) heterogenous frequencies.
In all cases we found that in optimized networks the large
frequencies are localized to hubs and frequencies of
neighboring oscillators are negatively correlated.
Although the theoretic approach developed herein is
valid for systems given by Eq. (1), extension to more
general models such as Landau-Stuart [28], Winfree [29]
and chaotic oscillators remains an outstanding problem.
One promising area stems from Kuramoto’s phase reduction methods, which give Eq. (1) as an approximation of the
dynamics of weakly interacting limit-cycle oscillators [10].
Another exciting venue of research would be the optimization of other dynamical patterns such as multistability,
hysteresis, and/or explosive synchronization, none of
which was observed here (despite the sharp transitions
seen in Figs. 1 and 2), but they can potentially arise under
more general dynamics.
Finally, we compare our results on heterogeneous
oscillators to the well-developed theory regarding identical
oscillators [30], for which the synchronizability of a network is given by the ratio λN =λ2 of Laplacian eigenvalues
[31]—a result allowing for optimization to be independent
of the node dynamics [32]. In contrast, we find here that the
synchronization of a network of heterogeneous oscillators
depends on the full set of eigenvalues and eigenvectors of
the network Laplacian–much like the case of nearly
identical oscillators [33] and real-world experiments
[34]–and also how the network structure pairs with the
heterogeneity of node dynamics (here oscillator frequencies). A network that is easily synchronizable with identical
oscillators may have poor synchronization properties with
heterogeneous oscillators, and vice versa.
This work was funded in part by the James S. McDonnell
Foundation (P. S. S.), NSF Grant No. DMS-1127914
through the Statistical and Applied Mathematical
Sciences Institute (D. T.), ARO Grant No. W911NF-121-0276 (J. S.), and Simons Foundation Grant No. 318812
(J. S.).
*
skardals@gmail.com
dane.r.taylor@gmail.com
‡
sunj@clarkson.edu
[1] S. H. Strogatz, Sync: The Emerging Science of Spontaneous
Order (Hypernion, New York 2003).
[2] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences
(Cambridge University Press, Cambridge, England, 2003).
[3] S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes,
Rev. Mod. Phys. 80, 1275 (2008).
[4] A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and
C. Zhou, Phys. Rep. 469, 93 (2008).
†
week ending
3 OCTOBER 2014
[5] J. Buck, Q. Rev. Biol. 63, 265 (1988).
[6] L. Glass and M. C. Mackey, From Clocks to Chaos: The
Rhythms of Life (Princeton University Press, Princeton, NJ,
1988).
[7] A. E. Motter, S. A. Myers, M. Anghel, and T. Nishikawa,
Nat. Phys. 9, 191 (2013).
[8] S. H. Strogatz, D. M. Abrams, A. McRobie, B. Eckhardt,
and E. Ott, Nature (London) 438, 43 (2005).
[9] A. Prindle, P. Samayoa, I. Razinkov, T. Danino, L. S.
Tsimring, and J. Hasty, Nature (London) 481, 39 (2012).
[10] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, New York, 1984).
[11] H. Sakaguchi and Y. Kuramoto, Prog. Theor. Phys. 76, 576
(1986).
[12] Y. Moreno and A. F. Pacheco, Europhys. Lett. 68, 603
(2004).
[13] T. Ichinomiya, Phys. Rev. E 70, 026116 (2004).
[14] J. G. Restrepo, E. Ott, and B. R. Hunt, Phys. Rev. E 71,
036151 (2005).
[15] A. Arenas, A. Díaz-Guilera, and C. J. Peréz-Vicente, Phys.
Rev. Lett. 96, 114102 (2006).
[16] J. Gómez-Gardeñes, Y. Moreno, and A. Arenas, Phys. Rev.
Lett. 98, 034101 (2007).
[17] J. Gómez-Gardeñes, S. Gómez, A. Arenas, and Y. Moreno,
Phys. Rev. Lett. 106, 128701 (2011).
[18] P. S. Skardal and J. G. Restrepo, Phys. Rev. E 85, 016208
(2012).
[19] P. S. Skardal and J. G. Restrepo, Proceedings of the 2012
International Symposium on Nonlinear Theory and its
Applications, October 22–26, 2012, Palma, Mallorca,
Spain.
[20] P. S. Skardal, J. Sun, D. Taylor, and J. G. Restrepo,
Europhys. Lett. 101, 20001 (2013).
[21] F. Dörfler, M. Chertkov, and F. Bullo, Proc. Natl. Acad. Sci.
U.S.A. 110, 2005 (2013).
[22] A. Ben-Israel and T. N. E. Grenville, Generalized Inverses
(Springer, New York, 1974).
[23] M. Molloy.and B. Reed, Random Structure and Algorithms
6, 161 (1995).
[24] C. Zhan, G. Chen, and L. F. Yeung, Physica (Amsterdam)
389A, 1779 (2010).
[25] P. Erdős and A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 5,
17 (1960).
[26] M. Brede, Phys. Lett. A 372, 2618 (2008).
[27] L. Buzna, S. Lozano, and A. Díaz-Guilera, Phys. Rev. E 80,
066120 (2009).
[28] M. Rosenblum and A. Pikovsky, Phys. Rev. Lett. 98,
064101 (2007).
[29] A. T. Winfree, J. Theor. Biol. 16, 15 (1967).
[30] L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 80, 2109
(1998).
[31] M. Barahona and L. M. Pecora, Phys. Rev. Lett. 89, 054101
(2002).
[32] T. Nishikawa and A. E. Motter, Proc. Natl. Acad. Sci.
U.S.A. 107, 10342 (2010).
[33] J. Sun, E. M. Bollt, and T. Nishikawa, Europhys. Lett. 85,
60011 (2009).
[34] B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy,
and R. Roy, Phys. Rev. Lett. 107, 034102 (2011).
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